Download Reading Strategies 8-5

Name
Date
Class
Reading Strategies
LESSON
8-5
Use a Concept Map
Use the laws below when a triangle is not a right triangle.
Example
Use when: You are given ASA,
AAS, or SSA.
sin 60 ______
sin 80
______
a
Formula
Law of Sines
sin C
sin B ____
sin A ____
____
a
10 cm sin 60 a sin 80
c
b
10 cm
10
cm sin 60 a
___________
sin 80
a 8.8 cm
#
B
!
—
—
A
—
CM
"
Use when: You are given SAS or SSS.
2
2
2
2
2
2
2
2
2
Example
Law of Cosines
Formulas
2
b (8.8) 10 2 2(8.8)(10)cos 40
a b c 2bccos A
2
2
b 77.44 100 (176)cos 40
b a c 2accos B
2
b 177.44 (176)cos 40
c a b 2abcos C
2
2
b 42.6 cm
b 6.5 cm
In Exercises 1–3, write the law you should use given the following.
1. Two sides and the included angle
2. Two angles and the included side
3. Three sides
4. Find AB. Round to the nearest unit.
5. Find m⬔C. Round to the nearest degree.
!
!
—
"
CM
—
FT
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
"
#
42
CM
CM
#
Holt Geometry
Review for Mastery
LESSON
8-5
8-5
continued
For any 䉭ABC with side lengths a, b, and c that are
opposite angles A, B, and C, respectively,
2
B
c
A
a
2
a b c 2bc cos A,
2
2
2
b a c 2ac cos B,
2
2
2
c a b 2ab cos C.
Law of Sines and Law of Cosines
A vertical stone pillar stands on a slope that makes a 22 angle with the
horizontal. At a time of day when the angle of elevation of the sun is 62,
the stone pillar casts a shadow that is 20.5 meters long as measured
along the slope.
The Law of Cosines
2
Challenge
LESSON
Law of Sines and Law of Cosines
%
b
sun·s rays
C
pillar
!
$
shadow
20.5 m
Find HK. Round to the nearest tenth.
HK 2 HJ 2 JK 2 2(HJ)(JK) cos J
289 196 2(17)(14) cos 50
2
HK 179.0331 ft
H
Law of Cosines
17 ft
50°
Substitute the known values.
2
Simplify.
HK 13.4 ft
J
14 ft
You can use the Law of Sines and the Law of Cosines to solve triangles according
to the information you have.
62°
Use the Law of Sines if you know
3. Find m⬔EDA.
4. Find m⬔DAE.
5. Set up a Law of Sines proportion that you can
solve to find the height of the pillar.
20.5 m
28.1 m
X
7. Draw a diagram and label the parts.
9 cm
6
43°
W
F
10 cm
8
7
Y
7.0 cm
pillar
28.1 m
sun·s rays
58
8. m⬔R
9. AB
21 mi
T
B
R
62°
11 km
15 mi
95°
28°
A
16 km
S
45
shadow
C
8. Set up a trigonometric equation to solve
for the length of the shadow.
8.1 km
28.1 m
tan 62 ______
x
9. Find the length of the shadow to the
nearest tenth.
39
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
LESSON
8-5
001-062_Go08an_CRF_c08.indd 39
h
Follow the instructions in Exercises 7–9 to find how long a shadow a stone
pillar of the same height would cast at the same time of day if it were standing
on level ground instead of a slope.
7. m⬔X
E
sin 40
sin 28 ______
_______
6. Find the height of the pillar to the nearest tenth.
Find each measure. Round lengths to the nearest tenth and angle measures to
the nearest degree.
6. EF
䉭ABC, 䉭DAE, 䉭FCE, 䉭BDF
28
40
112
2. Find m⬔DEA.
Use the Law of Cosines if you know
#
&
1. Name the triangles in this diagram.
• two angle measures and any side length, or • two side lengths and the included angle
measure, or
• two side lengths and a nonincluded angle
measure
• three side lengths
D
22°
"
K
Find the square root of both sides.
Holt Geometry
Problem Solving
1. The map shows three earthquake centers
for one week in California. How far apart
were the earthquake centers at points
A and C ? Round to the nearest tenth.
8-5
Use a Concept Map
4/13/07 9:59:30 AM
Use the laws below when a triangle is not a right triangle.
2. A BMX track has a starting hill as shown
in the diagram. What is the length of the
hill, WY ? Round to the nearest tenth.
8
—
a
9
a
—
#
10
cm sin 60 a
___________
sin 80
a 8.8 cm
#
B
!
122°
—
CM
"
2
2
2
a b c 2bc cos A
2
b 77.44 100 (176)cos 40
2
b 177.44 (176)cos 40
2
2
2
c a b 2abcos C
—
M
Example
2
2
2
b (8.8) 10 2(8.8)(10)cos 40
2
2
2
b a c 2ac cos B
Choose the best answer. Use the following information and
diagram for Exercises 5 and 6.
2
A
Law of Cosines
Formulas
60°
To find the distance across a bay, a surveyor
locates points Q, R, and S as shown.
—
—
Use when: You are given SAS or SSS.
4. The coordinates of the vertices of 䉭HJK
are H(0, 4), J(5, 7), and K(9, 1). Find
the measure of ⬔H to the nearest degree.
23.3 mi
2
2
b 42.6 cm
b 6.5 cm
3
In Exercises 1–3, write the law you should use given the following.
5. What is QR to the nearest tenth?
A 8m
C 41.9 m
Law of Cosines
1. Two sides and the included angle
M
D 55.4 m
Law of Sines
2. Two angles and the included side
6. What is m⬔Q to the nearest degree?
F 43°
H 67°
10 cm
10 cm sin 60 a sin 80
c
b
3. The edges of a triangular cushion
measure 8 inches, 3 inches, and 6 inches.
What is the measure of the largest angle
of the cushion to the
nearest degree?
3AN$IEGO
G 49°
Law of Sines
sin C
sin A ____
sin B ____
____
32.9 ft
MI
B 35.2 m
sin 60 ______
sin 80
______
Formula
FT
!
Example
Use when: You are given ASA,
AAS, or SSA.
7
"
/CEANSIDE
Holt Geometry
Reading Strategies
LESSON
Law of Sines and Law of Cosines
14.9 m
40
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
Law of Cosines
3. Three sides
1
J 107°
4. Find AB. Round to the nearest unit.
5. Find m⬔C. Round to the nearest degree.
!
!
7. Two angles of a triangle measure 56° and
77°. The side opposite the 56° angle is
29 cm long. What is the measure of the
shortest side? Round to the nearest tenth.
A 23.4 cm
B 25.6 cm
C 32.9 cm
D 34.1 cm
8. Which is the best estimate for the
perimeter of a triangle if two sides
measure 7 inches and 10 inches, and
the included angle between the two sides
is 82°?
F 11.4 in.
H 28.4 in.
G 12.2 in.
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
41
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
—
"
CM
—
FT
"
#
8 ft
CM
CM
#
55
J 39.9 in.
Holt Geometry
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
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Holt Geometry
Holt Geometry